Improved estimates for the number of non-negative integer matrices with given row and column sums

TL;DR

This paper introduces a new approximation method for counting non-negative integer matrices with specified row and column sums, offering improved accuracy and efficiency over existing methods, and provides code for implementation.

Contribution

It presents a novel approximation based on non-integer column considerations and an enhanced numerical method for counting and sampling matrices.

Findings

The new approximation is as accurate or better than existing methods.

The method runs in linear time relative to matrix size.

Code for the methods is provided.

Abstract

The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We also use this new estimate as the starting point for an improved numerical method for either counting or sampling matrices using sequential importance sampling. Code implementing our methods is provided.